www

commit 40556076097a869a13ca70690cbece82f8ff02c9
parent ef2de270fd03cc7ce7a91990055fa7642b7af72c
Author: Georges Dupéron <georges.duperon@gmail.com>
Date:   Thu, 22 Jun 2017 12:45:33 +0200

Accidentally had an unnecessary WXME file.

Diffstat:
M
scribblings/tr.scrbl
 | 
7908
++++++-------------------------------------------------------------------------

1 file changed, 604 insertions(+), 7304 deletions(-)
diff --git a/scribblings/tr.scrbl b/scribblings/tr.scrbl
@@ -1,7306 +1,605 @@
-#reader(lib"read.ss""wxme")WXME0108 ## 
-#|
-   This file uses the GRacket editor format.
-   Open this file in DrRacket version 6.9.0.4 or later to read it.
+#lang scribble/manual
 
-   Most likely, it was created by saving a program in DrRacket,
-   and it probably contains a program with non-text elements
-   (such as images or comment boxes).
+@require["util.rkt"
+         (for-label (only-meta-in 0 typed/racket))
+         scribble/example
+         racket/string]
+@(use-mathjax)
 
-            http://racket-lang.org/
-|#
- 25 7 #"wxtext\0"
-3 1 6 #"wxtab\0"
-1 1 8 #"wximage\0"
-2 0 8 #"wxmedia\0"
-4 1 34 #"(lib \"syntax-browser.ss\" \"mrlib\")\0"
-1 0 36 #"(lib \"cache-image-snip.ss\" \"mrlib\")\0"
-1 0 68
-(
- #"((lib \"image-core.ss\" \"mrlib\") (lib \"image-core-wxme.rkt\" \"mr"
- #"lib\"))\0"
-) 1 0 16 #"drscheme:number\0"
-3 0 44 #"(lib \"number-snip.ss\" \"drscheme\" \"private\")\0"
-1 0 36 #"(lib \"comment-snip.ss\" \"framework\")\0"
-1 0 93
-(
- #"((lib \"collapsed-snipclass.ss\" \"framework\") (lib \"collapsed-sni"
- #"pclass-wxme.ss\" \"framework\"))\0"
-) 0 0 43 #"(lib \"collapsed-snipclass.ss\" \"framework\")\0"
-0 0 19 #"drscheme:sexp-snip\0"
-0 0 29 #"drscheme:bindings-snipclass%\0"
-1 0 101
-(
- #"((lib \"ellipsis-snip.rkt\" \"drracket\" \"private\") (lib \"ellipsi"
- #"s-snip-wxme.rkt\" \"drracket\" \"private\"))\0"
-) 2 0 88
-(
- #"((lib \"pict-snip.rkt\" \"drracket\" \"private\") (lib \"pict-snip.r"
- #"kt\" \"drracket\" \"private\"))\0"
-) 0 0 55
-#"((lib \"snip.rkt\" \"pict\") (lib \"snip-wxme.rkt\" \"pict\"))\0"
-1 0 34 #"(lib \"bullet-snip.rkt\" \"browser\")\0"
-0 0 25 #"(lib \"matrix.ss\" \"htdp\")\0"
-1 0 22 #"drscheme:lambda-snip%\0"
-1 0 29 #"drclickable-string-snipclass\0"
-0 0 26 #"drracket:spacer-snipclass\0"
-0 0 57
-#"(lib \"hrule-snip.rkt\" \"macro-debugger\" \"syntax-browser\")\0"
-1 0 18 #"java-comment-box%\0"
-1 0 23 #"java-interactions-box%\0"
-1 0 1 6 #"wxloc\0"
-          0 0 70 0 1 #"\0"
-0 75 1 #"\0"
-0 12 90 -1 90 -1 3 -1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 255 255 255 1 -1 0 9
-#"Standard\0"
-0 75 10 #"Monospace\0"
-0 13 90 -1 90 -1 3 -1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 255 255 255 1 -1 2 1
-#"\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 -1 -1 2 24
-#"framework:default-color\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 -1 -1 2 1
-#"\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 1 1 1 150 0 150 0 0 0 -1 -1 2 15
-#"text:ports out\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 93 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 150 0 150 0 0 0 -1 -1 2 1
-#"\0"
-0 -1 1 #"\0"
-1.0 0 -1 -1 93 -1 -1 -1 0 0 0 0 0 0 0 0 0 1.0 1.0 1.0 255 0 0 0 0 0 -1
--1 2 15 #"text:ports err\0"
-0 -1 1 #"\0"
-1 0 -1 -1 93 -1 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 255 0 0 0 0 0 -1 -1 2 1
-#"\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 175 0 0 0 -1 -1 2 17
-#"text:ports value\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 93 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 0 0 175 0 0 0 -1 -1 2 1
-#"\0"
-0 -1 1 #"\0"
-1.0 0 92 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 1.0 1.0 1.0 34 139 34 0 0 0 -1
--1 2 27 #"Matching Parenthesis Style\0"
-0 -1 1 #"\0"
-1.0 0 92 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 1.0 1.0 1.0 34 139 34 0 0 0 -1
--1 2 1 #"\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 93 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 38 38 128 0 0 0 -1 -1 2 37
-#"framework:syntax-color:scheme:symbol\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 93 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 38 38 128 0 0 0 -1 -1 2 38
-#"framework:syntax-color:scheme:keyword\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 93 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 38 38 128 0 0 0 -1 -1 2 1
-#"\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 93 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 194 116 31 0 0 0 -1 -1 2
-38 #"framework:syntax-color:scheme:comment\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 93 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 194 116 31 0 0 0 -1 -1 2 1
-#"\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 93 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 41 128 38 0 0 0 -1 -1 2 37
-#"framework:syntax-color:scheme:string\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 93 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 41 128 38 0 0 0 -1 -1 2 35
-#"framework:syntax-color:scheme:text\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 93 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 41 128 38 0 0 0 -1 -1 2 39
-#"framework:syntax-color:scheme:constant\0"
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-#"\0"
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-#"framework:syntax-color:scheme:hash-colon-keyword\0"
-0 -1 1 #"\0"
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-#"framework:syntax-color:scheme:parenthesis\0"
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-#"\0"
-0 -1 1 #"\0"
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-#"framework:syntax-color:scheme:error\0"
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-#"\0"
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-#"framework:syntax-color:scheme:other\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 93 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 -1 -1 2 16
-#"Misspelled Text\0"
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-#"\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 93 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 81 112 203 0 0 0 -1 -1 2
-38 #"drracket:check-syntax:lexically-bound\0"
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-#"\0"
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-1 0 -1 -1 -1 93 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 178 34 34 0 0 0 -1 -1 2 28
-#"drracket:check-syntax:set!d\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 93 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 178 34 34 0 0 0 -1 -1 2 37
-#"drracket:check-syntax:unused-require\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 93 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 255 0 0 0 0 0 -1 -1 2 36
-#"drracket:check-syntax:free-variable\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 93 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 255 0 0 0 0 0 -1 -1 2 1
-#"\0"
-0 -1 1 #"\0"
-1 0 -1 -1 -1 93 -1 -1 0 1 0 0 0 0 0 0 0 1 1 1 68 0 203 0 0 0 -1 -1 2 31
-#"drracket:check-syntax:imported\0"
-0 -1 1 #"\0"
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-#"drracket:check-syntax:my-obligation-style-pref\0"
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-#"\0"
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-#"drracket:check-syntax:their-obligation-style-pref\0"
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-#"drracket:check-syntax:unk-obligation-style-pref\0"
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-#"\0"
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-#"\0"
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-41 #"profj:syntax-colors:scheme:block-comment\0"
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-#"\0"
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-#"profj:syntax-colors:scheme:keyword\0"
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-#"\0"
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-#"profj:syntax-colors:scheme:prim-type\0"
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-#"\0"
-0 -1 1 #"\0"
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-#"\0"
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-#"profj:syntax-colors:scheme:string\0"
-0 -1 1 #"\0"
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-#"profj:syntax-colors:scheme:literal\0"
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-#"profj:syntax-colors:scheme:error\0"
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-#"profj:syntax-colors:scheme:default\0"
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-#"plt:module-language:test-coverage-on\0"
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-          0 6971 0 28 3 21 #"#lang scribble/manual"
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-0 0 14 3 6 #"racket"
-0 0 24 3 1 #"["
-0 0 14 3 3 #"Bar"
-0 0 24 3 1 #"]"
-0 0 20 3 4 #" are"
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #"  "
-0 0 20 3 29 #"mutually recursive. The type "
-0 0 24 3 1 #"@"
-0 0 14 3 6 #"racket"
-0 0 24 3 1 #"["
-0 0 14 3 3 #"Foo"
-0 0 24 3 1 #"]"
-0 0 20 3 9 #" matches "
-0 0 20 3 6 #"lists "
-0 0 20 3 22 #"with an even number of"
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #"  "
-0 0 20 3 12 #"alternating "
-0 0 24 3 1 #"@"
-0 0 14 3 6 #"racket"
-0 0 24 3 1 #"["
-0 0 14 3 7 #"Integer"
-0 0 24 3 1 #"]"
-0 0 20 3 5 #" and "
-0 0 24 3 1 #"@"
-0 0 14 3 6 #"racket"
-0 0 24 3 1 #"["
-0 0 14 3 6 #"String"
-0 0 24 3 1 #"]"
-0 0 20 3 27 #" elements, starting with an"
-0 0 24 29 1 #"\n"
-0 0 24 3 3 #"  @"
-0 0 14 3 6 #"racket"
-0 0 24 3 1 #"["
-0 0 14 3 7 #"Integer"
-0 0 24 3 1 #"]"
-0 0 20 3 1 #","
-0 0 24 29 1 #"\n"
-0 0 24 29 1 #"\n"
-0 0 24 3 3 #"  @"
-0 0 14 3 11 #"racketblock"
-0 0 24 3 1 #"["
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #" ("
-0 0 15 3 11 #"define-type"
-0 0 24 3 1 #" "
-0 0 14 3 3 #"Foo"
-0 0 24 3 2 #" ("
-0 0 14 3 6 #"Pairof"
-0 0 24 3 1 #" "
-0 0 14 3 7 #"Integer"
-0 0 24 3 1 #" "
-0 0 14 3 3 #"Bar"
-0 0 24 3 2 #"))"
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #" ("
-0 0 15 3 11 #"define-type"
-0 0 24 3 1 #" "
-0 0 14 3 3 #"Bar"
-0 0 24 3 2 #" ("
-0 0 14 3 6 #"Pairof"
-0 0 24 3 1 #" "
-0 0 14 3 6 #"String"
-0 0 24 3 2 #" ("
-0 0 14 3 1 #"U"
-0 0 24 3 1 #" "
-0 0 14 3 3 #"Foo"
-0 0 24 3 1 #" "
-0 0 14 3 4 #"Null"
-0 0 24 3 4 #")))]"
-0 0 24 29 1 #"\n"
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #"  "
-0 0 20 3 5 #"This "
-0 0 20 3 5 #"same "
-0 0 20 3 5 #"type "
-0 0 20 3 6 #"could "
-0 0 20 3 17 #"alternatively be "
-0 0 20 3 18 #"defined using the "
-0 0 24 3 1 #"@"
-0 0 14 3 6 #"racket"
-0 0 24 3 1 #"["
-0 0 14 3 3 #"Rec"
-0 0 24 3 1 #"]"
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #"  "
-0 0 20 3 23 #"operator. The notation "
-0 0 24 3 1 #"@"
-0 0 14 3 6 #"racket"
-0 0 24 3 2 #"[("
-0 0 14 3 3 #"Rec"
-0 0 24 3 1 #" "
-0 0 14 3 1 #"R"
-0 0 24 3 1 #" "
-0 0 14 3 1 #"T"
-0 0 24 3 2 #")]"
-0 0 20 3 1 #" "
-0 0 20 3 16 #"builds the type "
-0 0 24 3 1 #"@"
-0 0 14 3 6 #"racket"
-0 0 24 3 1 #"["
-0 0 14 3 1 #"T"
-0 0 24 3 1 #"]"
-0 0 20 3 7 #", where"
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #"  "
-0 0 20 3 15 #"occurrences of "
-0 0 24 3 1 #"@"
-0 0 14 3 6 #"racket"
-0 0 24 3 1 #"["
-0 0 14 3 1 #"R"
-0 0 24 3 1 #"]"
-0 0 20 3 1 #" "
-0 0 20 3 4 #"are "
-0 0 20 3 39 #"interpreted as recursive occurrences of"
-0 0 24 29 1 #"\n"
-0 0 24 3 3 #"  @"
-0 0 14 3 6 #"racket"
-0 0 24 3 1 #"["
-0 0 14 3 1 #"T"
-0 0 24 3 1 #"]"
-0 0 20 3 8 #" itself."
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #"  "
-0 0 24 29 1 #"\n"
-0 0 24 3 3 #"  @"
-0 0 14 3 11 #"racketblock"
-0 0 24 3 1 #"["
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #" ("
-0 0 14 3 3 #"Rec"
-0 0 24 3 1 #" "
-0 0 14 3 1 #"R"
-0 0 24 29 1 #"\n"
-0 0 24 3 7 #"      ("
-0 0 14 3 6 #"Pairof"
-0 0 24 3 1 #" "
-0 0 14 3 7 #"Integer"
-0 0 24 29 1 #"\n"
-0 0 24 3 15 #"              ("
-0 0 14 3 6 #"Pairof"
-0 0 24 3 1 #" "
-0 0 14 3 6 #"String"
-0 0 24 29 1 #"\n"
-0 0 24 3 23 #"                      ("
-0 0 14 3 1 #"U"
-0 0 24 3 1 #" "
-0 0 14 3 1 #"R"
-0 0 24 3 1 #" "
-0 0 14 3 4 #"Null"
-0 0 24 3 6 #"))))]}"
-0 0 24 29 1 #"\n"
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #" @"
-0 0 14 3 8 #"asection"
-0 0 24 3 1 #"{"
-0 0 24 29 1 #"\n"
-0 0 24 3 3 #"  @"
-0 0 14 3 6 #"atitle"
-0 0 24 3 1 #"{"
-0 0 20 3 7 #"Classes"
-0 0 24 3 1 #"}"
-0 0 24 29 1 #"\n"
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #"  "
-0 0 20 3 3 #"The"
-0 0 20 3 1 #" "
-0 0 24 3 1 #"@"
-0 0 14 3 13 #"racketmodname"
-0 0 24 3 1 #"["
-0 0 14 3 12 #"racket/class"
-0 0 24 3 1 #"]"
-0 0 20 3 1 #" "
-0 0 20 3 6 #"module"
-0 0 20 3 1 #" "
-0 0 20 3 8 #"provides"
-0 0 20 3 30 #" an object-oriented system for"
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #"  "
-0 0 20 3 11 #"Racket. It "
-0 0 20 3 9 #"supports "
-0 0 20 3 4 #"the "
-0 0 20 3 14 #"definition of "
-0 0 20 3 8 #"classes "
-0 0 20 3 5 #"with "
-0 0 20 3 12 #"methods and "
-0 0 20 3 7 #"fields."
-0 0 24 29 1 #"\n"
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #"  "
-0 0 20 3 26 #"We will not describe this "
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #" }"
-0 0 24 29 1 #"\n"
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #" @"
-0 0 14 3 8 #"asection"
-0 0 24 3 1 #"{"
-0 0 24 29 1 #"\n"
-0 0 24 3 3 #"  @"
-0 0 14 3 6 #"atitle"
-0 0 24 3 1 #"{"
-0 0 20 3 17 #"Occurrence typing"
-0 0 24 3 1 #"}"
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #" }"
-0 0 24 29 1 #"\n"
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #" @"
-0 0 14 3 8 #"asection"
-0 0 24 3 1 #"{"
-0 0 24 29 1 #"\n"
-0 0 24 3 3 #"  @"
-0 0 14 3 6 #"atitle"
-0 0 24 3 1 #"{"
-0 0 20 3 21 #"Global type inference"
-0 0 24 3 1 #"}"
-0 0 24 29 1 #"\n"
-0 0 24 3 2 #" }"
-0 0 24 29 1 #"\n"
-0 0 24 3 1 #"}"
-0           0
+@(define tr-eval (make-eval-factory '(typed/racket)))
+
+@title[#:style (with-html5 manual-doc-style)
+       #:version (version-text)
+       #:tag "tr-chap"]{@|Typedracket|}
+
+We start this section with some history: Lisp, @emph{the} language with lots
+of parentheses, shortly following Fortran as one of the first high-level
+programming languages, was initially designed between 1956 and 1958, and
+subsequently implemented@~cite["McCarthyHistoryLisp"]. Dialects of Lisp
+generally support a variety of programming paradigms, including (but not
+limited to) functional programming and object-oriented programming (e.g. via
+CLOS, the Common Lisp Object System). One of the the most proeminent aspects
+of Lisp is homoiconicity, the fact that programs and data structures look the
+same. This enables programs to easily manipulate other programs, and led to
+the extensive use of macros. Uses of macros usually look like function
+applications, but, instead of invoking a target function at run-time, a macro
+will perform some computation at compile-time, and expand to some new code,
+which is injected as a replacement of the macro's use.
+
+The two main dialects of Lisp are Common Lisp and Scheme. Scheme follows a
+minimalist philosophy, where a small core is
+standardised@~cite["r5rs" "r6rs" "r7rs"] and subsequently extended via macros
+and additional function definitions.
+
+Racket, formerly named PLT Scheme, started as a Scheme implementation. Racket
+evolved, and the Racket Manifesto@~cite["racketmanifesto"] presents it as a
+``programming-language programming language'', a language which helps with the
+creation of small linguistic extensions as well as entirely new languages. The
+Racket ecosystem features many languages covering many paradigms:
+
+@itemlist[
+ @item{The @racketmodname[racket/base] language is a full-featured programming
+  language which mostly encourages functional programming.}
+ @item{@racketmodname[racket/class] implements
+  @seclink["classes" #:doc '(lib "scribblings/guide/guide.scrbl")]{an
+   object-oriented system}, implemented atop @racketmodname[racket/base] using
+  macros, and can be used along with the rest of the @racketmodname[racket/base]
+  language.}
+ @item{@racketmodname[racklog] is a logic programming language in the style of
+  prolog. The Racket ecosystem also includes an implementation of
+  @racketmodname[datalog].}
+ @item{@seclink["top" #:doc '(lib "scribblings/scribble/scribble.scrbl")]{
+   Scribble} can be seen as an alternative to @|LaTeX|, and is used to create
+  the @seclink["top" #:doc '(lib "scribblings/main/start.scrbl")]{Racket
+   documentation}. It also supports literate programming, by embedding chunks of
+  code in the document which are then aggregated together. This thesis is
+  in fact written using Scribble.}
+ @item{@racketmodname[slideshow] is a @deftech{DSL} (domain-specific language)
+  for the creation of presentations, and can be thought as an alternative to
+  Beamer and SliTeX.}
+ @item{@racketmodname[r5rs] and @racketmodname[r6rs] are implementations of
+  the corresponding scheme standards.}
+ @item{@seclink["top" #:doc '(lib "redex/redex.scrbl")]{Redex} is a
+  @usetech{DSL} which allows the specification of reduction
+  semantics for programming languages. It features tools to explore and test
+  the defined semantics.}
+ @item{@|Typedracket|@~cite["tobin-hochstadt_design_2008"
+                            "tobin-hochstadt_typed_2010"] is a typed variant of
+  the main @racketmodname[racket] language. It is implemented as a macro which
+  takes over the whole body of the program. That macro fully expands all other
+  macros in the program, and then typechecks the expanded program.}
+ @item{@seclink["top" #:doc '(lib "turnstile/scribblings/turnstile.scrbl")]{
+   @|Turnstile|} allows the creation of new typed languages. It takes a
+  different approach when compared to @|typedracket|, and threads the type
+  information through assignments and special forms, in order to be able to
+  typecheck the program during expansion, instead of doing so afterwards.}]
+
+In the remainder of this section, we will present the features of
+@|typedracket|'s type system, and then present formal semantics for a subset
+of those, namely the part which is relevant to our work.
+@other-doc['(lib "typed-racket/scribblings/ts-guide.scrbl")] and
+@other-doc['(lib "typed-racket/scribblings/ts-reference.scrbl")] provide good
+documentation for programmers who desire to use @|typedracket|; we will
+therefore keep our overview succinct and gloss over most details.
+
+@asection{
+ @atitle{Overview of Typed Racket's type system}
+
+ @asection{
+  @atitle{Simple primitive types}
+   
+  @Typedracket has types matching Racket's baggage of primitive values:
+  @racket[Number], @racket[Boolean], @racket[Char], @racket[String],
+  @racket[Void]@note{The @racket[Void] type contains only a single value,
+   @racket[#,(void)], and is equivalent to the @racketid[void] type in
+   @|C-language|. It is the equivalent of @racketid[unit] of @CAML and
+   @|haskell|, and is often used as the return type of functions which perform
+   side-effects. It should not be confused with @racket[Nothing], the bottom
+   type which is not inhabited by any value, and is similar to the type of
+   @|haskell|'s @racketid[undefined]. @racket[Nothing] can be used for example
+   as the type of functions which never return — in that way it is similar to
+   @|C-language|'s @tt["__attribute__ ((__noreturn__))"].} and so on.
+
+  @examples[#:label #f #:eval (tr-eval)
+            (ann #true Boolean)
+            243
+            "Hello world"
+            #\c
+            (code:comment "The void function produces the void value")
+            (code:comment "Void values on their own are not printed,")
+            (code:comment "so we place it in a list to make it visible.")
+            (list (void))]
+  
+  For numbers, @|typedracket| offers a ``numeric tower'' of
+  partially-overlapping types: @racket[Positive-Integer] is a subtype of
+  @racket[Integer], which is itself a subtype of @racket[Number]. @racket[Zero],
+  the type containing only the number 0, is a both a subtype of
+  @racket[Nonnegative-Integer] (numbers ≥ 0) and of @racket[Nonpositive-Integer]
+  (numbers ≤ 0).
+
+  @|Typedracket| also includes a singleton type for each primitive value of
+  these types: we already mentioned @racket[Zero], which is an alias of the
+  @racket[0] type. Every number, character, string and boolean value can be used
+  as a type, which is only inhabited by the same number, character, string or
+  boolean value. For example, @racket[243] belongs to the singleton type
+  @racket[243], which is a subtype of @racket[Positive-Integer].
+
+  @examples[#:label #f #:eval (tr-eval)
+            0
+            (ann 243 243)
+            #t]}
+
+ @asection{
+  @atitle{Pairs and lists}
+
+  Pairs are the central data structure of most Lisp dialects. They are used to
+  build linked lists of pairs, terminated by @racket['()], the null element. The
+  null element has the type @racket[Null], while the pairs which build the list
+  have the type @racket[(Pairof _A _B)], where @racketid[_A] and @racketid[_B]
+  are replaced by the actual types for the first and second elements of the
+  pair. For example, the pair built using @racket[(cons 729 #true)], which
+  contains @racket[729] as its first element, and @racket[#true] as its second
+  element, has the type @racket[(Pairof Number Boolean)], or using the most
+  precise singleton types, @racket[(Pairof 729 #true)].
+
+  @examples[#:label #f #:eval (tr-eval)
+            (cons 729 #true)
+            '(729 . #true)]
+
+  Heterogeneous linked lists of fixed length can be given a precise type by
+  nesting the same number of pairs at the type level. For example, the list
+  built with @racket[(list 81 #true 'hello)] has the type
+  @racket[(List Number Boolean Symbol)], which is a shorthand for the type
+  @racket[(Pairof Number (Pairof Boolean (Pairof Symbol Null)))]. Lists in
+  @|typedracket| can thus be seen as the equivalent of a chain of nested
+  2-tuples in languages like @|CAML| or @|haskell|. The analog in
+  object-oriented languages with support for generics would be a class
+  @tt["Pair<A, B>"], where the generic type argument @racketid[B] could be
+  instantiated by another instance of @tt["Pair"], and so on.
+
+  @examples[#:label #f #:eval (tr-eval)
+            (cons 81 (cons #true (cons 'hello null)))
+            (ann (list 81 #true 'hello)
+                 (Pairof Number (Pairof Boolean (Pairof Symbol Null))))]
+
+  The type of variable-length homogeneous linked lists can be described using
+  the @racket[Listof] type operator. The type @racket[(Listof Integer)] is
+  equivalent to @racket[(Rec R (U (Pairof Integer R) Null))]. The @racket[Rec]
+  type operator describes @seclink["tr-presentation-recursive-types"]{recursive
+   types}, and @racket[U] describes @seclink["tr-presentation-unions"]{unions}.
+  Both of these features are described below, for now we will simply say that
+  the previously given type is a recursive type @racket[R], which can be a
+  @racket[(Pairof Integer R)] or @racket[Null] (to terminate the linked list).
+
+  @examples[#:label #f #:eval (tr-eval)
+            (ann (range 0 5) (Listof Number))]}
+
+ @asection{
+  @atitle{Symbols}
+
+  Another of Racket's primitive datatypes is symbols. Symbols are interned
+  strings: two occurrences of a symbol produce values which are pointer-equal if
+  the symbols are equal (i.e. they represent the same string)@note{This is true
+   with the exception of symbols created with @racket[gensym] and the like.
+   @racket[gensym] produces a fresh symbol which is not interned, and therefore
+   different from all existing symbols, and different from all symbols created
+   in the future.}.
+
+  @|Typedracket| includes the @racket[Symbol] type, to which all symbols
+  belong. Additionally, there is a singleton type for each symbol: the type
+  @racket['foo] is only inhabited by the symbol @racket['foo].
+
+  @examples[#:label #f #:eval (tr-eval)
+            'foo]
+
+  Singleton types containing symbols can be seen as similar to constructors
+  without arguments in @|CAML| and @|haskell|, and as globally unique enum
+  values in object-oriented languages. The main difference resides in the scope
+  of the declaration: two constructor declarations with identical names in two
+  separate files will usually give distinct types and values. Similarly, when
+  using the ``type-safe enum'' design pattern, two otherwise identical
+  declarations of an enum will yield objects of different types. In contrast,
+  two uses of an interned symbols in Racket and @|typedracket| will produce
+  identical values and types. A way of seeing this is that symbols are similar
+  to constructors (in the functional programming sense) or enums which are
+  implicitly declared globally.
+
+  @examples[#:label #f #:eval (tr-eval)
+            (module m1 typed/racket
+              (define sym1 'foo)
+              (provide sym1))
+            (module m2 typed/racket
+              (define sym2 'foo)
+              (provide sym2))
+            (require 'm1 'm2)
+            (code:comment "The tow independent uses of 'foo are identical:")
+            (eq? sym1 sym2)]
+ }
+  
+ @asection{
+  @atitle[#:tag "tr-presentation-unions"]{Unions}
+
+  These singleton types may not seem very useful on their own. They can however
+  be combined together with union types, which are built using the @racket[U]
+  type operator.
+  
+  The union type @racket[(U 0 1 2)] is inhabited by the values @racket[0],
+  @racket[1] and @racket[2], and by no other value. The @racket[Boolean] type is
+  actually defined as @racket[(U #true #false)], i.e. the union of the singleton
+  types containing the @racket[#true] and @racket[#false] values, respectively.
+  The @racket[Nothing] type, which is not inhabited by any value, is defined as
+  the empty union @racket[(U)]. The type @racket[Any] is the top type, i.e. it
+  is a super-type of all other types, and can be seen as a large union including
+  all other types, including those which will be declared later or in other
+  units of code.
+
+  Unions of symbols are similar to variants which contain zero-argument
+  constructors, in @|CAML| or @|haskell|.
+
+  @examples[#:label #f #:eval (tr-eval)
+            (define v : (U 'foo 'bar) 'foo)
+            v
+            (set! v 'bar)
+            v
+            (code:comment "This throws an error at compile-time:")
+            (eval:error (set! v 'oops))]
+
+  A union such as @racket[(U 'ca (List 'cb Number) (List 'cc String Symbol))]
+  can be seen as roughly the equivalent of a variant with three constructors,
+  @racketid[ca], @racket[cb] and @racketid[cc], where the first has no
+  arguments, the second has one argument (a @racket[Number]), and the third has
+  two arguments (a @racket[String] and a @racket[Symbol]).
+
+  The main difference is that a symbol can be used as parts of several unions,
+  e.g. @racket[(U 'a 'b)] and @racket[(U 'b 'c)], while constructors can often
+  only be part of the variant used to declare them. Unions of symbols are in
+  this sense closer to @|CAML|'s so-called polymorphic
+  variants@~cite["minskyRealWorldOCaml"] than to regular variants.
+  
+  @examples[#:label #f #:eval (tr-eval)
+            (define-type my-variant (U 'ca
+                                       (List 'cb Number)
+                                       (List 'cc String Symbol)))
+            (define v₁ : my-variant 'ca)
+            (define v₂ : my-variant (list 'cb 2187))
+            (define v3 : my-variant (list 'cc "Hello" 'world))]
+
+  Finally, it is possible to mix different sorts of types within the same
+  union: the type @racket[(U 0 #true 'other)] is inhabited by the number
+  @racket[0], the boolean @racket[#true], and the symbol @racket['other].
+  Translating such an union to a language like @|CAML| could be done by
+  explicitly tagging each case of the union with a distinct constructor.
+
+  Implementation-wise, all values in the so-called ``untyped'' version of
+  Racket are tagged: a few bits within the value's representation are reserved
+  and used to encode the value's type. When considering the target of a pointer
+  in memory, Racket is therefore able to determine if the pointed-to value is a
+  number, boolean, string, symbol and so on. Typed Racket preserves these
+  run-time tags. They can then be used to detect the concrete type of a value
+  when its static type is a union. This detection is done simply by using
+  Racket's predicates: @racket[number?], @racket[string?], @racket[symbol?]
+  etc.}
+
+ @asection{
+  @atitle{Intersections}
+
+  Intersections are the converse of unions: instead of allowing a mixture of
+  values of different types, an intersection type, described using the
+  @racket[∩] type operator, only allows values which belong to all types.
+
+  The intersection type @racket[(∩ Nonnegative-Integer Nonpositive-Integer)] is
+  the singleton type @racket[0]. The intersection of @racket[(U 'a 'b 'c)] and
+  @racket[(U 'b 'c 'd)] will be @racket[(U 'b 'c)], as @racket['b] and
+  @racket['c] belong to both unions.
+
+  @examples[
+ #:label #f #:eval (tr-eval)
+ (code:comment ":type shows the given type, or a simplified version of it")
+ (:type (∩ (U 'a 'b 'c) (U 'b 'c 'd)))]
+  
+  @|Typedracket| is able to reduce some intersections such as those given above
+  at compile-time. However, in some cases, it is forced to keep the intersection
+  type as-is. For example, structs (@seclink["tr-presentation-structs"]{
+   describled below} can, using special properties, impersonate functions. This
+  mechanism is similar to PHP's @tt["__invoke"], the ability to overload
+  @tt["operator()"] in @|CPP|. @|Typedracket| does not handle these properties
+  (yet), and therefore cannot determine whether a given struct type also
+  impersonates a function or not. This means that the intersection
+  @racket[(∩ s (→ Number String))], where @racket[s] is a struct type, cannot be
+  reduced to @racket[Nothing], because @|typedracket| cannot determine whether
+  the struct @racket[s] can act as a function or not.
+
+  Another situation where @|typedracket| cannot reduce the intersection is when
+  intersecting two function types (@seclink["tr-presentation-functions"]{
+   presented below}).
+
+  @racketblock[
+ (∩ (→ Number String) (→ Number Symbol))
+ (∩ (→ Number String) (→ Boolean String))]
+
+  The first intersection seems like could be simplified to
+  @racket[(→ Number String) (→ Number Symbol)], and the second one could be
+  simplified to @racket[(→ (U Number Boolean) String)], however the equivalence
+  between these types has not been implemented (yet) in @|typedracket|, so we do
+  not rely on them. Note that this issue is not a soundness issue: it only
+  prevents passing values  types to which they belong in principle, but it
+  cannot be exploited to assign a value to a variable with an incompatible type.
+
+  Finally, when some types are intersected with a polymorphic type variable,
+  the intersection cannot be computed until the polymorphic type is
+  instantiated.
+  
+  When @|typedracket| is able to perform a simplification, occurrences of
+  @racket[Nothing] (the bottom type) propagate outwards in some cases, pairs and
+  struct types which contain @racket[Nothing] as one of their elements being
+  collapsed to @racket[Nothing]. This propagation of @racket[Nothing] starts
+  from occurrences of @racket[Nothing] in the parts of the resulting type which
+  are traversed by the intersection operator. It collapses the containing pairs
+  and struct types to @racket[Nothing], moving outwards until the @racket[∩]
+  operator itself is reached. In principle, the propagation could go on past
+  that point, but this is not implemented yet in @|typedracket|@note{See
+   @hyperlink["https://github.com/racket/typed-racket/issues/552"]{Issue #552}
+   on @|typedracket|'s GitHub repository for more details on what prevents
+   implementing a more aggressive propagation of @racket[Nothing].}.
+
+  The type @racket[(∩ 'a 'b)] therefore gets simplified to @racket[Nothing],
+  and the type @racket[(∩ (Pairof 'a 'x) (Pairof 'b 'x))] also simplifies to
+  @racket[Nothing] (@|typedracket| initially pushes the intersection down the
+  pairs, so that the type first becomes @racket[(Pairof (∩ 'a 'b) (∩ 'x 'x))],
+  which is simplified to @racket[(Pairof Nothing 'x)], and the occurrence of
+  @racket[Nothing] propagates outwards). However, if the user directly specifies
+  the type @racket[(Pairof (∩ 'a 'b) Integer)], it is simplified to
+  @racket[(Pairof Nothing Integer)], but the @racket[Nothing] does not propagate
+  outwards beyond the initial use of @racket[∩].
+
+  @examples[#:label #f #:eval (tr-eval)
+            (:type (∩ 'a 'b))
+            (:type (∩ (Pairof 'a 'x) (Pairof 'b 'x)))
+            (:type (Pairof (∩ 'a 'b) Integer))]
+
+  A simple workaround exists: the outer type, which could be collapsed to
+  @racket[Nothing], can be intersected again with a type of the same shape. The
+  outer intersection will traverse both types (the desired one and the
+  ``shape''), and propagate the leftover @racket[Nothing] further out.
+
+  @examples[#:label #f #:eval (tr-eval)
+            (:type (Pairof (∩ 'a 'b) Integer))
+            (:type (∩ (Pairof (∩ 'a 'b) Integer)
+                      (Pairof Any Any)))]
+
+  These intersections are not very interesting on their own, as in most cases
+  it is possible to express the resulting simplified type without using the
+  intersection operator. They become more useful when mixed with polymorphic
+  types: intersecting a polymorphic type variable with another type can be used
+  to restrict the actual values that may be used. The type @racket[(∩ A T)],
+  where @racket[A] is a polymorphic type variable and @racket[T] is a type
+  defined elsewhere, is equivalent to the use of bounded type parameters in
+  @|java| or @|csharp|. In @|csharp|, for example, the type @racket[(∩ A T)]
+  would be written using an @tt["where A : T"] clause.}
+ 
+ @asection{
+  @atitle[#:tag "tr-presentation-structs"]{Structs}
+
+  Racket also supports @racket[struct]s, which are mappings from fields to
+  values. A struct is further distinguished by its struct type: instances of two
+  struct types with the same name and fields, declared in separate files, can be
+  differentiated using the predicates associated with these structs. Structs in
+  Racket can be seen as the analog of classes containing only fields (but no
+  methods) in @csharp or @|java|. Such classes are sometimes called ``Plain Old
+  Data (POD) Objects''. Structs belong to a single-inheritance hierarchy:
+  instances of the descendents of a struct type are recognised by their
+  ancestor's predicate. When a struct inherits from another, it includes its
+  parent's fields, and can add extra fields of its own.
+
+  Each struct declaration within a @|typedracket| program additionally declares
+  corresponding type.
+
+  @examples[#:label #f #:eval (tr-eval)
+            (struct parent ([field₁ : (Pairof String Symbol)])
+              #:transparent)
+            (struct s parent ([field₂ : Integer]
+                              [field₃ : Symbol])
+              #:transparent)
+            (s (cons "x" 'y) 123 'z)]
+
+  In @|typedracket|, structs can have polymorphic type arguments, which can be
+  used inside the types of the struct's fields.
+
+  @examples[#:label #f #:eval (tr-eval)
+            (struct (A B) poly-s ([field₁ : (Pairof A B)]
+                                  [field₂ : Integer]
+                                  [field₃ : B])
+              #:transparent)
+            (poly-s (cons "x" 'y) 123 'z)]
+
+  Racket further supports
+  @tech[#:doc '(lib "scribblings/reference/reference.scrbl")]{struct type
+   properties}, which can be seen as a limited form of method definitions for a
+  struct, thereby making them closer to real objects. The same struct type
+  property can be implemented by many structs, and the declaration of a struct
+  type property is therefore roughly equivalent to the declaration of an
+  interface with a single method.
+
+  Struct type properties are often considered a low-level mechanism in Racket.
+  Among other things, a struct type property can only be used to define a single
+  property at a time. When multiple ``methods'' have to be defined at once (for
+  example, when defining the @racket[prop:equal+hash] property, which requires
+  the definition of an equality comparison function, and two hashing functions),
+  these can be grouped together in a list of functions, which is then used as
+  the property's value.
+  ``@seclink["struct-generics"
+             #:doc '(lib "scribblings/reference/reference.scrbl")]{
+   Generic interfaces}'' are a higher-level feature, which among other things
+  allow the definition of multiple ``methods'' as part of a single generic
+  interface, and offers a friendlier API for specifying the ``generic
+  interface'' itself (i.e. what Object Oriented languages call an interfece), as
+  and for specifying the implementation of said interface.
+
+  @|Typedracket| unfortunately offers no support for struct type properties and
+  generic interfaces for now. It is impossible to assert that a struct
+  implements a given property at the type level, and it is also for example not
+  possible to describe the type of a function accepting any struct implementing
+  a given property or generic interface. Finally, no type checks are performed
+  on the body of functions bound to such properties, and to check verifies that
+  a function implementation with the right signature is supplied to a given
+  property. Since struct type properties and generics cannot be used in a
+  type-safe way for now, we refrain from using these features, and only use them
+  to implement some very common properties@note{We built a thin macro wrapper
+   which allows typechecking the implementation and signature of the functions
+   bound to these two properties.}: @racket[prop:custom-write] which is the
+  equivalent of @|java|'s @tt["void toString()"], and @racket[prop:equal+hash]
+  which is equivalent to @|java|'s @tt["boolean equals(Object o)"] and
+  @tt["int hashCode()"].
+
+ }
+ 
+ @asection{
+  @atitle[#:tag "tr-presentation-functions"]{Functions}
+   
+  @|Typedracket| supports rich function types, to support some of the flexible
+  use patterns allowed by Racket.
+
+  The simple function type below indicates that the function expects two
+  arguments (an integer and a string), and returns a boolean:
+
+  @racketblock[(→ Integer String Boolean)]
+
+  We note that unlike @|haskell| and @|CAML| functions, Racket functions are
+  not implicitly curried. To express the corresponding curried function type,
+  one would write:
+
+  @racketblock[(→ Integer (→ String Boolean))]
+
+  A function may additionally accept optional positional arguments, and keyword
+  (i.e. named) arguments, both mandatory and optional:
+
+  @racketblock[
+ (code:comment "Mandatory string, optional integer and boolean arguments:")
+ (->* (String) (Integer Boolean) Boolean)
+ (code:comment "Mandatory keyword arguments:")
+ (→ #:size Integer #:str String Boolean)
+ (code:comment "Mandatory #:str, optional #:size and #:opt:")
+ (->* (#:str String) (#:size Integer #:opt Boolean) Boolean)]
+
+  Furthermore, functions in Racket accept a catch-all ``rest'' argument, which
+  allows for the definition of variadic functions. Typed racket also allows
+  expressing this at the type level, as long as the arguments covered by the
+  ``rest'' clause all have the same type:
+
+  @racketblock[
+ (code:comment "The function accepts one integer and any number of strings:")
+ (-> Integer String * Boolean)
+ (code:comment "Same thing with an optional symbol inbetween: ")
+ (->* (Integer) (Symbol) #:rest String Boolean)]
+
+  One of @|typedracket|'s main goals is to be able to typecheck idiomatic
+  Racket programs. Such programs may include functions whose return type depends
+  on the values of the input arguments. Similarly, @racket[case-lambda] can be
+  used to create lambda functions which dispatch to multiple behaviours based on
+  the number of arguments passed to the function.
+
+  @|Typedracket| provides the @racket[case→] type operator, which can be used to
+  describe the type of these functions:
+
+  @racketblock[
+ (code:comment "Allows 1 or 3 arguments, with the same return type.")
+ (case→ (→ Integer Boolean)
+        (→ Integer String Symbol Boolean))
+ (code:comment "A similar type based on optional arguments allows 1, 2 or 3")
+ (code:comment " arguments in contrast:")
+ (->* (Integer) (String Symbol) Boolean)
+ (code:comment "The output type can depend on the input type:")
+ (case→ (→ Integer Boolean)
+        (→ String Symbol))
+ (code:comment "Both features (arity and dependent output type) can be mixed")
+ (case→ (→ Integer Boolean)
+        (→ Integer String (Listof Boolean)))]
+
+  Another important feature, which can be found in the type system of most
+  functional programming languages, and most object-oriented languages, is
+  parametric polymorphism. @|Typedracket| allows the definition of polymorphic
+  structs, as detailed above, as well as polymorphic functions. For example, the
+  function @racket[cons] can be considered as a polymorphic function with two
+  polymorphic type arguments @racket[A] and @racket[B], which takes an argument
+  of type @racket[A], an argument of type @racket[B], and returns a pair of
+  @racket[A] and @racket[B].
+
+  @racketblock[(∀ (A B) (→ A B (Pairof A B)))]
+
+  @htodo{Something on which types can be inferred and which can't (for now).}
+
+  Finally, predicates (functions whose results can be interpreted as booleans)
+  can be used to gain information about the type of their argument, depending on
+  the result. The type of a predicate can include positive and negative filters,
+  indicated with @racket[#:+] and @racket[#:-], respectively. The type of the
+  @racket[string?] predicate is:
+
+  @racketblock[(→ Any Boolean : #:+ String #:- (! String))]
+
+  In this notation, the positive filter @racket[#:+ String] indicates that when
+  the predicate returns @racket[#true], the argument is known to be a
+  @racket[String]. Conversely, when the predicate exits with @racket[#false],
+  the negative filter @racket[#:- (! String)] indicates that the input could not
+  (@racket[!]) possibly have been a string. The information gained this way
+  allows regular conditionals based on arbitrary predicates to work like
+  pattern-matching:
+
+  @examples[#:label #f #:eval (tr-eval)
+            (define (f [x : (U String Number Symbol)])
+              (if (string? x)
+                  (code:comment "x is known to be a String here:")
+                  (ann x String)
+                  (code:comment "x is known to be a Number or a Symbol here:")
+                  (ann x (U Number Symbol))))]
+
+  The propositions do not necessarily need to refer to the value as a whole,
+  and can instead give information about a sub-part of the value. Right now, the
+  user interface for specifying paths can only target the left and right members
+  of @racket[cons] pairs, recursively. Internally, @|typedracket| supports
+  richer paths, and the type inference can produce filters which give
+  information about individual structure fields, or about the result of forced
+  promises, for example.}
+
+ @asection{
+  @atitle[#:tag "tr-presentation-recursive-types"]{Recursive types}
+   
+  @|Typedracket| allows recursive types, both via (possibly mutually-recursive)
+  named declarations, and via the @racket[Rec] type operator.
+
+  In the following examples, the types @racket[Foo] and @racket[Bar] are
+  mutually recursive. The type @racket[Foo] matches lists with an even number of
+  alternating @racket[Integer] and @racket[String] elements, starting with an
+  @racket[Integer],
+
+  @racketblock[
+ (define-type Foo (Pairof Integer Bar))
+ (define-type Bar (Pairof String (U Foo Null)))]
+
+  This same type could alternatively be defined using the @racket[Rec]
+  operator. The notation @racket[(Rec R T)] builds the type @racket[T], where
+  occurrences of @racket[R] are interpreted as recursive occurrences of
+  @racket[T] itself.
+  
+  @racketblock[
+ (Rec R
+      (Pairof Integer
+              (Pairof String
+                      (U R Null))))]}
+
+ @asection{
+  @atitle{Classes}
+
+  The @racketmodname[racket/class] module provides an object-oriented system for
+  Racket. It supports the definition of classes with methods and fields.
+
+  We will not describe this 
+ }
+
+ @asection{
+  @atitle{Occurrence typing}
+ }
+
+ @asection{
+  @atitle{Global type inference}
+ }
+}
+\ No newline at end of file